1882.] Liquid Ellipsoid about an Axis. 209 



angular velocity whose components about the axes of the ellipsoid 

 are denoted by £, w, £; the velocity at any point of the liquid 

 relative to the ellipsoid is therefore zero. 



Any alteration of the angular velocity will however generate 

 motion in the liquid relative to the ellipsoid, such that if O t , 

 fi 2 , I2 3 denote the additional component angular velocities im- 

 parted to the ellipsoid, and if U, V, W denote the component 

 velocities of the liquid at the point (x, y, z) relative to the ellipsoid, 

 then 



2a 2 2a 2 



*-*+**#-?+*** 



Jr 26 2 W n 



V + c' 1 a z + b 2 3 

 2c 2 2c 2 



values independent of £, rj, £; and then, if o) 1 , co tJ , co a denote the 

 component angular velocities of the ellipsoid, 



°>i = n i + £ ®a = a a + Vj « 3 = ^3 + & 



In this manner any arbitrary motion of the liquid filling the 

 ellipsoid, subject to the condition of uniform vorticity throughout 

 the volume of the ellipsoid, may be supposed to have been 

 established. 



If M denote the mass of the liquid, and h l , h 2> h 3 the com- 

 ponents of angular momentum of the liquid about the axes of the 

 ellipsoid, then 



tf _i_ r* ( //j 2 — r' 2x 2 

 h=M b 



5 \V + c 



y^ + 4' 

 , r c 2 + a 2 f/c 2 -a 2 \ 2 ) 



as before ; and therefore if the liquid be suddenly frozen, the 

 component angular velocities of the ellipsoid would change to 



Oi + ft 



W+c'J 



c 2 — a' 

 c 2 + a' 1 



a 



